Since Drupal VM is built in a modular fashion, you can swap out the database engine and use PostgreSQL instead of MySQL (as long as the version of Drupal you're using supports it!).

To switch from MySQL to PostgreSQL, switch the drupalvm_database setting in your local config.yml to pgsql:

drupalvm_database: pgsql

For more PostgreSQL configuration options, see the README included with the geerlingguy.postgresql Ansible role.